<?xml version="1.0"?>
<div class="add">

<span class="kw">consider </span><font color="Maroon" title="c1">C</font> being   <a href="cat_6.html#V1" title="CAT_6:attr.1">strict</a>   <a href="cat_7.html#V6" title="CAT_7:attr.6">preorder</a>  <a href="cat_6.html#NM2" title="CAT_6:NM.2">category</a><span class="kw"> such that </span><br/><a NAME="E2:18"/><span class="lab"><font color="Green" title="E17">A1</font></span>: 
 <a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 <span class="kw">and </span><br/><a NAME="E3:18"/><span class="lab"><font color="Green" title="E18">A2</font></span>: 
 for <font color="Olive" title="b1">o1</font>, <font color="Olive" title="b2">o2</font> being   <a href="cat_6.html#NM3" title="CAT_6:NM.3">Object</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b1">o1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">o2</font> holds <br/> <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Olive" title="b1">o1</font>,<font color="Olive" title="b2">o2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">o1</font>,<font color="Olive" title="b2">o2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">and </span><br/><a NAME="E4:18"/><span class="lab"><font color="Green" title="E19">A3</font></span>: 
 <a href="cat_7.html#K5" title="CAT_7:func.5">RelOb</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> 3
 <span class="kw">and </span><br/><a NAME="E5:18"/><span class="lab"><font color="Green" title="E20">A4</font></span>: 
 <a href="cat_6.html#K1" title="CAT_6:func.1">Mor</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">o1</font>,<font color="Olive" title="b2">o2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">o1</font>, <font color="Olive" title="b2">o2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of 3 : <font color="Olive" title="b1">o1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">o2</font> </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_7.html#T37" onmouseover="rs('cat_7/T37')" onmouseout="rh()">CAT_7:37</a></span>;<br/>
<a NAME="E6:18"/><span class="lab"><font color="Green" title="E21">A5</font></span>: 
<font color="Maroon" title="c1">C</font> is 3 <a href="cat_7.html#V7" title="CAT_7:attr.7">-ordered</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:18"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="ref" href="wellord1.html#T38" onmouseover="rs('wellord1/T38')" onmouseout="rh()">WELLORD1:38</a>, <a class="ref" href="cat_7.html#D14" onmouseover="rs('cat_7/D14')" onmouseout="rh()">CAT_7:def 14</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c2">F</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C</font>,<span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 3</span>)</span>, <font color="Maroon" title="c3">G</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 3</span>)</span>,<font color="Maroon" title="c1">C</font><span class="kw"> such that </span><br/><a NAME="E8:18"/><span class="lab"><font color="Green" title="E22">A6</font></span>: 
( <font color="Maroon" title="c2">F</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c3">G</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  )
 <span class="kw">and </span><br/><a NAME="E9:18"/><span class="lab"><font color="Green" title="E23">A7</font></span>: 
( <font color="Maroon" title="c3">G</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c2">F</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c2">F</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c3">G</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 3</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:18"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="ref" href="cat_7.html#T38" onmouseover="rs('cat_7/T38')" onmouseout="rh()">CAT_7:38</a>, <a class="ref" href="cat_6.html#D28" onmouseover="rs('cat_6/D28')" onmouseout="rh()">CAT_6:def 28</a></span>;<br/>
<a NAME="E10:18"/><span class="lab"><font color="Green" title="E24">A8</font></span>: 
(  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> 1 &amp;  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of 3 &amp; 1 is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of 3 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T49" onmouseover="rs('card_1/T49')" onmouseout="rh()">CARD_1:49</a>, <a class="ref" href="card_1.html#T51" onmouseover="rs('card_1/T51')" onmouseout="rh()">CARD_1:51</a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a>, <a class="ref" href="enumset1.html#D1" onmouseover="rs('enumset1/D1')" onmouseout="rh()">ENUMSET1:def 1</a></span>;<br/>
<a NAME="E11:18"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A9</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">o1</font>,<font color="Olive" title="b2">o2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">o1</font>, <font color="Olive" title="b2">o2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of 3 : <font color="Olive" title="b1">o1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">o2</font> </span> } </span> 
 
;<br/>
<a NAME="E12:18"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A10</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_6.html#K1" title="CAT_6:func.1">Mor</a> <font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:18"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">g1</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c1">C</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:18"><span class="lab"><font color="Green" title="E25">A9</font></span></a>, <a class="txt" href="#E5:18"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E14:18"/><span class="lab"><font color="Green" title="E27">A11</font></span>: 
 not <font color="Maroon" title="c1">C</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18"><span class="lab"><font color="Green" title="E17">A1</font></span></a></span>;<br/>
<a NAME="E15:18"/><span class="lab"><font color="Green" title="E28">A12</font></span>: 
 not <font color="Maroon" title="c4">g1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="18_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:18_1"/>
<font color="Maroon" title="c4">g1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:18_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">g1</font> is   <a href="cat_6.html#NM3" title="CAT_6:NM.3">Object</a> of <font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:18"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="ref" href="cat_6.html#T22" onmouseover="rs('cat_6/T22')" onmouseout="rh()">CAT_6:22</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:18_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18"><span class="lab"><font color="Green" title="E17">A1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">set </span><font color="Maroon" title="c5">f1</font> = <font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font>;<br/>
<a NAME="E16:18"/><span class="lab"><font color="Green" title="E29">A13</font></span>: 
( 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> 2 &amp; 1 is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of 3 &amp; 2 is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of 3 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T50" onmouseover="rs('card_1/T50')" onmouseout="rh()">CARD_1:50</a>, <a class="ref" href="card_1.html#T51" onmouseover="rs('card_1/T51')" onmouseout="rh()">CARD_1:51</a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a>, <a class="ref" href="enumset1.html#D1" onmouseover="rs('enumset1/D1')" onmouseout="rh()">ENUMSET1:def 1</a></span>;<br/>
<a NAME="E17:18"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A14</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">o1</font>,<font color="Olive" title="b2">o2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">o1</font>, <font color="Olive" title="b2">o2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of 3 : <font color="Olive" title="b1">o1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">o2</font> </span> } </span> 
 
;<br/>
<a NAME="E18:18"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A15</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_6.html#K1" title="CAT_6:func.1">Mor</a> <font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:18"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">g2</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c1">C</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:18"><span class="lab"><font color="Green" title="E30">A14</font></span></a>, <a class="txt" href="#E5:18"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E20:18"/><span class="lab"><font color="Green" title="E32">A16</font></span>: 
 not <font color="Maroon" title="c6">g2</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="18_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:18_2"/>
<font color="Maroon" title="c6">g2</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:18_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">g2</font> is   <a href="cat_6.html#NM3" title="CAT_6:NM.3">Object</a> of <font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:18"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="ref" href="cat_6.html#T22" onmouseover="rs('cat_6/T22')" onmouseout="rh()">CAT_6:22</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:18_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18"><span class="lab"><font color="Green" title="E17">A1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">set </span><font color="Maroon" title="c7">f2</font> = <font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font>;<br/>
<a NAME="E21:18"/><span class="lab"><font color="Green" title="E33">A17</font></span>: 
(  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> 2 &amp;  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of 3 &amp; 2 is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of 3 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T50" onmouseover="rs('card_1/T50')" onmouseout="rh()">CARD_1:50</a>, <a class="ref" href="card_1.html#T51" onmouseover="rs('card_1/T51')" onmouseout="rh()">CARD_1:51</a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a>, <a class="ref" href="enumset1.html#D1" onmouseover="rs('enumset1/D1')" onmouseout="rh()">ENUMSET1:def 1</a></span>;<br/>
<a NAME="E22:18"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A18</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">o1</font>,<font color="Olive" title="b2">o2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">o1</font>, <font color="Olive" title="b2">o2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of 3 : <font color="Olive" title="b1">o1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">o2</font> </span> } </span> 
 
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">g3</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c1">C</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:18"><span class="lab"><font color="Green" title="E34">A18</font></span></a>, <a class="txt" href="#E5:18"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">f3</font> = <font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c8">g3</font>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">f2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 3</span>)</span> st <br/>(  not <font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp;  not <font color="Olive" title="b1">f2</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp;  <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Olive" title="b1">f2</font> &amp;  <a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 3</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Olive" title="b1">f2</font></span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> &amp;  <a href="cat_6.html#K1" title="CAT_6:func.1">Mor</a> <span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 3</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="enumset1.html#K4" title="ENUMSET1:func.4">{</a><span class="default"><span class="p2">(<span class="default"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Olive" title="b1">f2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span>,<font color="Olive" title="b1">f2</font>,<span class="p2">(<span class="default"><font color="Olive" title="b1">f2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span></span><a href="enumset1.html#K4" title="ENUMSET1:func.4">}</a></span> &amp;  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span>, <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span>, <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Olive" title="b1">f2</font>,<font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font>,<font color="Olive" title="b1">f2</font>,<font color="Olive" title="b1">f2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span> <a href="zfmisc_1.html#R4" title="ZFMISC_1:pred.4">are_mutually_distinct</a>  )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp;  not <font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp;  <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span> &amp;  <a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 3</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span></span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> &amp;  <a href="cat_6.html#K1" title="CAT_6:func.1">Mor</a> <span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 3</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="enumset1.html#K4" title="ENUMSET1:func.4">{</a><span class="default"><span class="p2">(<span class="default"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span></span><a href="enumset1.html#K4" title="ENUMSET1:func.4">}</a></span> &amp;  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span>, <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span>, <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span>,<font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span> <a href="zfmisc_1.html#R4" title="ZFMISC_1:pred.4">are_mutually_distinct</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E24:18"/><span class="lab"><font color="Green" title="E35">A19</font></span>: 
 not <font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a> 
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp;  <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span> &amp;  <a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 3</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span></span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> &amp;  <a href="cat_6.html#K1" title="CAT_6:func.1">Mor</a> <span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 3</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="enumset1.html#K4" title="ENUMSET1:func.4">{</a><span class="default"><span class="p2">(<span class="default"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span></span><a href="enumset1.html#K4" title="ENUMSET1:func.4">}</a></span> &amp;  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span>, <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span>, <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span>,<font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span> <a href="zfmisc_1.html#R4" title="ZFMISC_1:pred.4">are_mutually_distinct</a>  )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="18_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:18_3"/><span class="lab"><font color="Green" title="E36">A20</font></span>: 
<font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:18_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:18"><span class="lab"><font color="Green" title="E26">A10</font></span></a>, <a class="ref" href="cat_6.html#D1" onmouseover="rs('cat_6/D1')" onmouseout="rh()">CAT_6:def 1</a></span>;<br/>
<a NAME="E3:18_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T18" onmouseover="rs('funct_1/T18')" onmouseout="rh()">FUNCT_1:18</a></span>;<br/>
<a NAME="E4:18_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A21</font></span>: 
<span class="p1">(<span class="default"><a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D4" onmouseover="rs('struct_0/D4')" onmouseout="rh()">STRUCT_0:def 4</a></span>;<br/>
<a NAME="E5:18_3"/>
<font color="Maroon" title="c3">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_3"><span class="lab"><font color="Green" title="E36">A20</font></span></a>, <a class="ref" href="cat_6.html#D22" onmouseover="rs('cat_6/D22')" onmouseout="rh()">CAT_6:def 22</a>, <a class="txt" href="#E8:18"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="ref" href="cat_6.html#D25" onmouseover="rs('cat_6/D25')" onmouseout="rh()">CAT_6:def 25</a></span>;<br/>
<a NAME="E6:18_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:18"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="txt" href="#E14:18"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="ref" href="cat_6.html#T34" onmouseover="rs('cat_6/T34')" onmouseout="rh()">CAT_6:34</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:18_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:18"><span class="lab"><font color="Green" title="E28">A12</font></span></a>, <a class="txt" href="#E9:18"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="txt" href="#E14:18"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="txt" href="#E4:18_3"><span class="lab"><font color="Green" title="E37">A21</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E25:18"/><span class="lab"><font color="Green" title="E36">A22</font></span>: 
 not <font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a> 
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span> &amp;  <a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 3</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span></span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> &amp;  <a href="cat_6.html#K1" title="CAT_6:func.1">Mor</a> <span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 3</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="enumset1.html#K4" title="ENUMSET1:func.4">{</a><span class="default"><span class="p2">(<span class="default"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span></span><a href="enumset1.html#K4" title="ENUMSET1:func.4">}</a></span> &amp;  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span>, <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span>, <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span>,<font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span> <a href="zfmisc_1.html#R4" title="ZFMISC_1:pred.4">are_mutually_distinct</a>  )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="18_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:18_4"/><span class="lab"><font color="Green" title="E37">A23</font></span>: 
<font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:18_4"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:18"><span class="lab"><font color="Green" title="E31">A15</font></span></a>, <a class="ref" href="cat_6.html#D1" onmouseover="rs('cat_6/D1')" onmouseout="rh()">CAT_6:def 1</a></span>;<br/>
<a NAME="E3:18_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T18" onmouseover="rs('funct_1/T18')" onmouseout="rh()">FUNCT_1:18</a></span>;<br/>
<a NAME="E4:18_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A24</font></span>: 
<span class="p1">(<span class="default"><a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D4" onmouseover="rs('struct_0/D4')" onmouseout="rh()">STRUCT_0:def 4</a></span>;<br/>
<a NAME="E5:18_4"/>
<font color="Maroon" title="c3">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_4"><span class="lab"><font color="Green" title="E37">A23</font></span></a>, <a class="ref" href="cat_6.html#D22" onmouseover="rs('cat_6/D22')" onmouseout="rh()">CAT_6:def 22</a>, <a class="txt" href="#E8:18"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="ref" href="cat_6.html#D25" onmouseover="rs('cat_6/D25')" onmouseout="rh()">CAT_6:def 25</a></span>;<br/>
<a NAME="E6:18_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:18"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="txt" href="#E14:18"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="ref" href="cat_6.html#T34" onmouseover="rs('cat_6/T34')" onmouseout="rh()">CAT_6:34</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:18_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:18"><span class="lab"><font color="Green" title="E32">A16</font></span></a>, <a class="txt" href="#E9:18"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="txt" href="#E14:18"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="txt" href="#E4:18_4"><span class="lab"><font color="Green" title="E38">A24</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">reconsider </span><font color="Maroon" title="c10">o0</font> =  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  as   <a href="cat_6.html#NM3" title="CAT_6:NM.3">Object</a> of <font color="Maroon" title="c1">C</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="card_1.html#T51" onmouseover="rs('card_1/T51')" onmouseout="rh()">CARD_1:51</a>, <a class="ref" href="enumset1.html#D1" onmouseover="rs('enumset1/D1')" onmouseout="rh()">ENUMSET1:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">o1</font> = 1 as   <a href="cat_6.html#NM3" title="CAT_6:NM.3">Object</a> of <font color="Maroon" title="c1">C</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="card_1.html#T51" onmouseover="rs('card_1/T51')" onmouseout="rh()">CARD_1:51</a>, <a class="ref" href="enumset1.html#D1" onmouseover="rs('enumset1/D1')" onmouseout="rh()">ENUMSET1:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">o2</font> = 2 as   <a href="cat_6.html#NM3" title="CAT_6:NM.3">Object</a> of <font color="Maroon" title="c1">C</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="card_1.html#T51" onmouseover="rs('card_1/T51')" onmouseout="rh()">CARD_1:51</a>, <a class="ref" href="enumset1.html#D1" onmouseover="rs('enumset1/D1')" onmouseout="rh()">ENUMSET1:def 1</a></span>;<br/>
<a NAME="E29:18"/><span class="lab"><font color="Green" title="E37">A25</font></span>: 
 not <font color="Maroon" title="c1">C</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18"><span class="lab"><font color="Green" title="E17">A1</font></span></a></span>;<br/>
<a NAME="E30:18"/>
 <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c10">o0</font>,<font color="Maroon" title="c11">o1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:18"><span class="lab"><font color="Green" title="E24">A8</font></span></a>, <a class="txt" href="#E3:18"><span class="lab"><font color="Green" title="E18">A2</font></span></a></span>;<br/>
<a NAME="E31:18"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A26</font></span>: 
<font color="Maroon" title="c4">g1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c10">o0</font>,<font color="Maroon" title="c11">o1</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<a NAME="E32:18"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A27</font></span>: 
(  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c4">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">o0</font> &amp;  <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Maroon" title="c4">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">o1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:18"><span class="lab"><font color="Green" title="E37">A25</font></span></a>, <a class="ref" href="cat_7.html#T20" onmouseover="rs('cat_7/T20')" onmouseout="rh()">CAT_7:20</a></span>;<br/>
<a NAME="E33:18"/><span class="lab"><font color="Green" title="E40">A28</font></span>: 
( <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c4">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">o0</font> &amp; <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Maroon" title="c4">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">o1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:18"><span class="lab"><font color="Green" title="E38">A26</font></span></a>, <a class="txt" href="#E29:18"><span class="lab"><font color="Green" title="E37">A25</font></span></a>, <a class="ref" href="cat_7.html#T20" onmouseover="rs('cat_7/T20')" onmouseout="rh()">CAT_7:20</a></span>;<br/>
<a NAME="E34:18"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A29</font></span>: 
(  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">o0</font> &amp;  <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">o1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:18"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="txt" href="#E29:18"><span class="lab"><font color="Green" title="E37">A25</font></span></a>, <a class="ref" href="cat_6.html#T32" onmouseover="rs('cat_6/T32')" onmouseout="rh()">CAT_6:32</a></span>;<br/>
<a NAME="E35:18"/>
 <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c11">o1</font>,<font color="Maroon" title="c12">o2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:18"><span class="lab"><font color="Green" title="E29">A13</font></span></a>, <a class="txt" href="#E3:18"><span class="lab"><font color="Green" title="E18">A2</font></span></a></span>;<br/>
<a NAME="E36:18"/><span class="kw">then </span>
<font color="Maroon" title="c6">g2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c11">o1</font>,<font color="Maroon" title="c12">o2</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<a NAME="E37:18"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A30</font></span>: 
(  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c6">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">o1</font> &amp;  <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Maroon" title="c6">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">o2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:18"><span class="lab"><font color="Green" title="E37">A25</font></span></a>, <a class="ref" href="cat_7.html#T20" onmouseover="rs('cat_7/T20')" onmouseout="rh()">CAT_7:20</a></span>;<br/>
<a NAME="E38:18"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A31</font></span>: 
(  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">o1</font> &amp;  <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">o2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:18"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="txt" href="#E29:18"><span class="lab"><font color="Green" title="E37">A25</font></span></a>, <a class="ref" href="cat_6.html#T32" onmouseover="rs('cat_6/T32')" onmouseout="rh()">CAT_6:32</a></span>;<br/>
<span class="kw">thus </span><a NAME="E39:18"/><span class="lab"><font color="Green" title="E44">A32</font></span>: 
 <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:18"><span class="lab"><font color="Green" title="E41">A29</font></span></a>, <a class="txt" href="#E37:18"><span class="lab"><font color="Green" title="E42">A30</font></span></a>, <a class="txt" href="#E8:18"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="txt" href="#E29:18"><span class="lab"><font color="Green" title="E37">A25</font></span></a>, <a class="ref" href="cat_6.html#T32" onmouseover="rs('cat_6/T32')" onmouseout="rh()">CAT_6:32</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 3</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span></span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> &amp;  <a href="cat_6.html#K1" title="CAT_6:func.1">Mor</a> <span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 3</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="enumset1.html#K4" title="ENUMSET1:func.4">{</a><span class="default"><span class="p2">(<span class="default"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span></span><a href="enumset1.html#K4" title="ENUMSET1:func.4">}</a></span> &amp;  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span>, <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span>, <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span>,<font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span> <a href="zfmisc_1.html#R4" title="ZFMISC_1:pred.4">are_mutually_distinct</a>  )</span><br/>

<a NAME="E40:18"/><span class="lab"><font color="Green" title="E45">A33</font></span>: 
<font color="Maroon" title="c6">g2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c4">g1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:18"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="txt" href="#E32:18"><span class="lab"><font color="Green" title="E39">A27</font></span></a>, <a class="txt" href="#E37:18"><span class="lab"><font color="Green" title="E42">A30</font></span></a>, <a class="ref" href="cat_7.html#T5" onmouseover="rs('cat_7/T5')" onmouseout="rh()">CAT_7:5</a></span>;<br/>
<a NAME="E41:18"/><span class="kw">then </span>
(  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c4">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c4">g1</font> &amp;  <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c4">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Maroon" title="c6">g2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_7.html#T4" onmouseover="rs('cat_7/T4')" onmouseout="rh()">CAT_7:4</a></span>;<br/>
<a NAME="E42:18"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A34</font></span>: 
<font color="Maroon" title="c6">g2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c4">g1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c10">o0</font>,<font color="Maroon" title="c12">o2</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:18"><span class="lab"><font color="Green" title="E42">A30</font></span></a>, <a class="txt" href="#E32:18"><span class="lab"><font color="Green" title="E39">A27</font></span></a>, <a class="txt" href="#E14:18"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="ref" href="cat_7.html#T20" onmouseover="rs('cat_7/T20')" onmouseout="rh()">CAT_7:20</a></span>;<br/>
<a NAME="E43:18"/><span class="lab"><font color="Green" title="E47">A35</font></span>: 
<font color="Maroon" title="c2">F</font> is  <a href="cat_6.html#V14" title="CAT_6:attr.14">multiplicative</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:18"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="ref" href="cat_6.html#D25" onmouseover="rs('cat_6/D25')" onmouseout="rh()">CAT_6:def 25</a></span>;<br/>
<a NAME="E44:18"/><span class="lab"><font color="Green" title="E48">A36</font></span>: 
 <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c10">o0</font>,<font color="Maroon" title="c12">o2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:18"><span class="lab"><font color="Green" title="E33">A17</font></span></a>, <a class="txt" href="#E3:18"><span class="lab"><font color="Green" title="E18">A2</font></span></a></span>;<br/>
<a NAME="E45:18"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A37</font></span>: 
<font color="Maroon" title="c8">g3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c4">g1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:18"><span class="lab"><font color="Green" title="E46">A34</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<span class="lab"><font color="Green" title="E50">A38</font></span>: <font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c8">g3</font> = 
<font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c4">g1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:18"><span class="lab"><font color="Green" title="E48">A36</font></span></a>, <a class="txt" href="#E42:18"><span class="lab"><font color="Green" title="E46">A34</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:18"><span class="lab"><font color="Green" title="E47">A35</font></span></a>, <a class="txt" href="#E40:18"><span class="lab"><font color="Green" title="E45">A33</font></span></a>, <a class="ref" href="cat_6.html#D23" onmouseover="rs('cat_6/D23')" onmouseout="rh()">CAT_6:def 23</a></span>
;<br/>
<a NAME="E47:18"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 3</span>)</span> iff <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span></span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="18_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 3</span>)</span> iff <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span></span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> )</span><br/>

<div><span class="kw">hereby </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span></span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> implies <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 3</span>)</span> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:18_6_1"/><span class="lab"><font color="Green" title="E51">A39</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 3</span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span></span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">o</font> = <font color="Maroon" title="c13">x</font> as   <a href="cat_6.html#NM3" title="CAT_6:NM.3">Object</a> of <span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 3</span>)</span> ;<br/><a NAME="E3:18_6_1"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_6.html#K1" title="CAT_6:func.1">Mor</a> <span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 3</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_6_1"><span class="lab"><font color="Green" title="E51">A39</font></span></a></span>;<br/><a NAME="E4:18_6_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A40</font></span>: 
<font color="Maroon" title="c14">o</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 3</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D1" onmouseover="rs('cat_6/D1')" onmouseout="rh()">CAT_6:def 1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c15">f</font> = <font color="Maroon" title="c14">o</font> as   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 3</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_6_1"><span class="lab"><font color="Green" title="E51">A39</font></span></a></span>;<br/><a NAME="E6:18_6_1"/><span class="lab"><font color="Green" title="E53">A41</font></span>: 
<font color="Maroon" title="c3">G</font> is  <a href="cat_6.html#V13" title="CAT_6:attr.13">identity-preserving</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:18"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="ref" href="cat_6.html#D25" onmouseover="rs('cat_6/D25')" onmouseout="rh()">CAT_6:def 25</a></span>;<br/><span class="lab"><font color="Green" title="E54">A42</font></span>: <font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c3">G</font></span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:18"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="ref" href="cat_6.html#T34" onmouseover="rs('cat_6/T34')" onmouseout="rh()">CAT_6:34</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 3</span>)</span></span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:18"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="ref" href="struct_0.html#D4" onmouseover="rs('struct_0/D4')" onmouseout="rh()">STRUCT_0:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 3</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">o</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>
<br/>.= 
<font color="Maroon" title="c15">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:18_6_1"><span class="lab"><font color="Green" title="E52">A40</font></span></a>, <a class="ref" href="funct_1.html#T18" onmouseover="rs('funct_1/T18')" onmouseout="rh()">FUNCT_1:18</a></span>
;<br/><a NAME="E8:18_6_1"/>
<font color="Maroon" title="c3">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:18_6_1"><span class="lab"><font color="Green" title="E53">A41</font></span></a>, <a class="ref" href="cat_6.html#T22" onmouseover="rs('cat_6/T22')" onmouseout="rh()">CAT_6:22</a>, <a class="ref" href="cat_6.html#D22" onmouseover="rs('cat_6/D22')" onmouseout="rh()">CAT_6:def 22</a></span>;<br/><a NAME="E9:18_6_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f1</font> where <font color="Olive" title="b1">f1</font> is   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c1">C</font> : ( <font color="Olive" title="b1">f1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Olive" title="b1">f1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_6.html#K1" title="CAT_6:func.1">Mor</a> <font color="Maroon" title="c1">C</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:18"><span class="lab"><font color="Green" title="E27">A11</font></span></a></span>;<br/><a NAME="E10:18_6_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1,2</span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="card_1.html#T51" onmouseover="rs('card_1/T51')" onmouseout="rh()">CARD_1:51</a>, <a class="ref" href="cat_6.html#D17" onmouseover="rs('cat_6/D17')" onmouseout="rh()">CAT_6:def 17</a></span>;<br/><a NAME="E11:18_6_1"/><span class="kw">then </span>
( <font color="Maroon" title="c3">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">o0</font> or <font color="Maroon" title="c3">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">o1</font> or <font color="Maroon" title="c3">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">o2</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#D1" onmouseover="rs('enumset1/D1')" onmouseout="rh()">ENUMSET1:def 1</a></span>;<br/><a NAME="E12:18_6_1"/><span class="kw">then </span>
( <font color="Maroon" title="c15">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">o0</font> or <font color="Maroon" title="c15">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">o1</font> or <font color="Maroon" title="c15">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">o2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:18_6_1"><span class="lab"><font color="Green" title="E54">A42</font></span></a>, <a class="txt" href="#E29:18"><span class="lab"><font color="Green" title="E37">A25</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>;<br/><span class="kw">hence </span><a NAME="E13:18_6_1"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span></span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:18"><span class="lab"><font color="Green" title="E41">A29</font></span></a>, <a class="txt" href="#E38:18"><span class="lab"><font color="Green" title="E43">A31</font></span></a>, <a class="ref" href="enumset1.html#D1" onmouseover="rs('enumset1/D1')" onmouseout="rh()">ENUMSET1:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:18_6"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span></span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 3</span>)</span></span><br/>

<a NAME="E3:18_6"/><span class="kw">then </span>
( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span> or <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span> or <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#D1" onmouseover="rs('enumset1/D1')" onmouseout="rh()">ENUMSET1:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:18_6"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 3</span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E48:18"/><span class="lab"><font color="Green" title="E51">A43</font></span>: 
 <a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 3</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span></span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T2" onmouseover="rs('tarski/T2')" onmouseout="rh()">TARSKI:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="cat_6.html#K1" title="CAT_6:func.1">Mor</a> <span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 3</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="enumset1.html#K4" title="ENUMSET1:func.4">{</a><span class="default"><span class="p2">(<span class="default"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span></span><a href="enumset1.html#K4" title="ENUMSET1:func.4">}</a></span> &amp;  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span>, <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span>, <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span>,<font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span> <a href="zfmisc_1.html#R4" title="ZFMISC_1:pred.4">are_mutually_distinct</a>  )</span><br/>

<a NAME="E49:18"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_6.html#K1" title="CAT_6:func.1">Mor</a> <span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 3</span>)</span> iff <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K4" title="ENUMSET1:func.4">{</a><span class="default"><span class="p2">(<span class="default"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span></span><a href="enumset1.html#K4" title="ENUMSET1:func.4">}</a></span> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="18_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_6.html#K1" title="CAT_6:func.1">Mor</a> <span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 3</span>)</span> iff <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K4" title="ENUMSET1:func.4">{</a><span class="default"><span class="p2">(<span class="default"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span></span><a href="enumset1.html#K4" title="ENUMSET1:func.4">}</a></span> )</span><br/>

<div><span class="kw">hereby </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K4" title="ENUMSET1:func.4">{</a><span class="default"><span class="p2">(<span class="default"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span></span><a href="enumset1.html#K4" title="ENUMSET1:func.4">}</a></span> implies <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_6.html#K1" title="CAT_6:func.1">Mor</a> <span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 3</span>)</span> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:18_7_1"/><span class="lab"><font color="Green" title="E52">A44</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_6.html#K1" title="CAT_6:func.1">Mor</a> <span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 3</span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K4" title="ENUMSET1:func.4">{</a><span class="default"><span class="p2">(<span class="default"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span></span><a href="enumset1.html#K4" title="ENUMSET1:func.4">}</a></span></span><br/><a NAME="E2:18_7_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A45</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 3</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D1" onmouseover="rs('cat_6/D1')" onmouseout="rh()">CAT_6:def 1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">f</font> = <font color="Maroon" title="c13">x</font> as   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 3</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_7_1"><span class="lab"><font color="Green" title="E52">A44</font></span></a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_7_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:18_7_1_1"/>
( <font color="Maroon" title="c14">f</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  or  not <font color="Maroon" title="c14">f</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_7_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:18_7_1_1_1"/>
<font color="Maroon" title="c14">f</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K4" title="ENUMSET1:func.4">{</a><span class="default"><span class="p2">(<span class="default"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span></span><a href="enumset1.html#K4" title="ENUMSET1:func.4">}</a></span></span><br/><div class="add"><a NAME="E2:18_7_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">f</font> is   <a href="cat_6.html#NM3" title="CAT_6:NM.3">Object</a> of <span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 3</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#T22" onmouseover="rs('cat_6/T22')" onmouseout="rh()">CAT_6:22</a></span>;<br/><a NAME="E3:18_7_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span> or <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span> or <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:18"><span class="lab"><font color="Green" title="E51">A43</font></span></a>, <a class="ref" href="enumset1.html#D1" onmouseover="rs('enumset1/D1')" onmouseout="rh()">ENUMSET1:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E4:18_7_1_1_1"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K4" title="ENUMSET1:func.4">{</a><span class="default"><span class="p2">(<span class="default"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span></span><a href="enumset1.html#K4" title="ENUMSET1:func.4">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#D4" onmouseover="rs('enumset1/D4')" onmouseout="rh()">ENUMSET1:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_7_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:18_7_1_1_2"/><span class="lab"><font color="Green" title="E54">A46</font></span>: 
 not <font color="Maroon" title="c14">f</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K4" title="ENUMSET1:func.4">{</a><span class="default"><span class="p2">(<span class="default"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span></span><a href="enumset1.html#K4" title="ENUMSET1:func.4">}</a></span></span><br/><div class="add"><a NAME="E2:18_7_1_1_2"/><span class="lab"><font color="Green" title="E55">A47</font></span>: 
<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 3</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18_7_1"><span class="lab"><font color="Green" title="E53">A45</font></span></a>, <a class="ref" href="funct_1.html#T18" onmouseover="rs('funct_1/T18')" onmouseout="rh()">FUNCT_1:18</a></span>;<br/><span class="lab"><font color="Green" title="E56">A48</font></span>: <font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c14">f</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c3">G</font></span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c14">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:18"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="ref" href="cat_6.html#T34" onmouseover="rs('cat_6/T34')" onmouseout="rh()">CAT_6:34</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 3</span>)</span></span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c14">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:18"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="ref" href="struct_0.html#D4" onmouseover="rs('struct_0/D4')" onmouseout="rh()">STRUCT_0:def 4</a></span>
<br/>.= 
<font color="Maroon" title="c14">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18_7_1_1_2"><span class="lab"><font color="Green" title="E55">A47</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>
;<br/><a NAME="E4:18_7_1_1_2"/>
 not <font color="Maroon" title="c3">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c14">f</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18_7_1_1_2"><span class="lab"><font color="Green" title="E56">A48</font></span></a>, <a class="txt" href="#E1:18_7_1_1_2"><span class="lab"><font color="Green" title="E54">A46</font></span></a>, <a class="ref" href="cat_6.html#D22" onmouseover="rs('cat_6/D22')" onmouseout="rh()">CAT_6:def 22</a>, <a class="txt" href="#E8:18"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="ref" href="cat_6.html#D25" onmouseover="rs('cat_6/D25')" onmouseout="rh()">CAT_6:def 25</a></span>;<br/><a NAME="E5:18_7_1_1_2"/><span class="kw">then </span>
 not <font color="Maroon" title="c3">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c14">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> 3
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="txt" href="#E14:18"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="ref" href="cat_6.html#T22" onmouseover="rs('cat_6/T22')" onmouseout="rh()">CAT_6:22</a></span>;<br/><a NAME="E6:18_7_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c14">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">o1</font>,<font color="Olive" title="b2">o2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">o1</font>, <font color="Olive" title="b2">o2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of 3 : <font color="Olive" title="b1">o1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">o2</font> </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:18"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">o1</font>, <font color="Maroon" title="c16">o2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of 3<span class="kw"> such that </span><br/><a NAME="E8:18_7_1_1_2"/><span class="lab"><font color="Green" title="E57">A49</font></span>: 
( <font color="Maroon" title="c3">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c14">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">o1</font>,<font color="Maroon" title="c16">o2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c15">o1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c16">o2</font> )
 ;<br/><a NAME="E9:18_7_1_1_2"/><span class="lab"><font color="Green" title="E58">A50</font></span>: 
( <font color="Maroon" title="c15">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c15">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c15">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T51" onmouseover="rs('card_1/T51')" onmouseout="rh()">CARD_1:51</a>, <a class="ref" href="enumset1.html#D1" onmouseover="rs('enumset1/D1')" onmouseout="rh()">ENUMSET1:def 1</a></span>;<br/><a NAME="E10:18_7_1_1_2"/>
( <font color="Maroon" title="c16">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c16">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c16">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T51" onmouseover="rs('card_1/T51')" onmouseout="rh()">CARD_1:51</a>, <a class="ref" href="enumset1.html#D1" onmouseover="rs('enumset1/D1')" onmouseout="rh()">ENUMSET1:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E11:18_7_1_1_2"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K4" title="ENUMSET1:func.4">{</a><span class="default"><span class="p2">(<span class="default"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span></span><a href="enumset1.html#K4" title="ENUMSET1:func.4">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18_7_1_1_2"><span class="lab"><font color="Green" title="E56">A48</font></span></a>, <a class="txt" href="#E46:18"><span class="lab"><font color="Green" title="E50">A38</font></span></a>, <a class="ref" href="enumset1.html#D4" onmouseover="rs('enumset1/D4')" onmouseout="rh()">ENUMSET1:def 4</a>, <a class="txt" href="#E8:18_7_1_1_2"><span class="lab"><font color="Green" title="E57">A49</font></span></a>, <a class="txt" href="#E9:18_7_1_1_2"><span class="lab"><font color="Green" title="E58">A50</font></span></a>, <a class="ref" href="card_1.html#T49" onmouseover="rs('card_1/T49')" onmouseout="rh()">CARD_1:49</a>, <a class="ref" href="card_1.html#T50" onmouseover="rs('card_1/T50')" onmouseout="rh()">CARD_1:50</a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:18_7"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K4" title="ENUMSET1:func.4">{</a><span class="default"><span class="p2">(<span class="default"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span></span><a href="enumset1.html#K4" title="ENUMSET1:func.4">}</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_6.html#K1" title="CAT_6:func.1">Mor</a> <span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 3</span>)</span></span><br/>

<a NAME="E3:18_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A51</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span></span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T13" onmouseover="rs('enumset1/T13')" onmouseout="rh()">ENUMSET1:13</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_7_2"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:18_7_2"/>
( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 3</span>)</span> or <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18_7"><span class="lab"><font color="Green" title="E52">A51</font></span></a>, <a class="txt" href="#E48:18"><span class="lab"><font color="Green" title="E51">A43</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_7_2_1"><span class="kw">suppose </span></a><a NAME="E1:18_7_2_1"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 3</span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_6.html#K1" title="CAT_6:func.1">Mor</a> <span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 3</span>)</span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:18_7_2_1"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_6.html#K1" title="CAT_6:func.1">Mor</a> <span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 3</span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_7_2_2"><span class="kw">suppose </span></a><a NAME="E1:18_7_2_2"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_6.html#K1" title="CAT_6:func.1">Mor</a> <span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 3</span>)</span></span><br/><div class="add"><a NAME="E2:18_7_2_2"/><span class="kw">then </span>
( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font> or <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font> or <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#D1" onmouseover="rs('enumset1/D1')" onmouseout="rh()">ENUMSET1:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:18_7_2_2"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_6.html#K1" title="CAT_6:func.1">Mor</a> <span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 3</span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E50:18"/>
 <a href="cat_6.html#K1" title="CAT_6:func.1">Mor</a> <span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 3</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="enumset1.html#K4" title="ENUMSET1:func.4">{</a><span class="default"><span class="p2">(<span class="default"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span></span><a href="enumset1.html#K4" title="ENUMSET1:func.4">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T2" onmouseover="rs('tarski/T2')" onmouseout="rh()">TARSKI:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span>, <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span>, <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span>,<font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span> <a href="zfmisc_1.html#R4" title="ZFMISC_1:pred.4">are_mutually_distinct</a> </span><br/>

<a NAME="E51:18"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T50" onmouseover="rs('card_1/T50')" onmouseout="rh()">CARD_1:50</a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/>
<a NAME="E52:18"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">o1</font>,<font color="Olive" title="b2">o2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">o1</font>, <font color="Olive" title="b2">o2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of 3 : <font color="Olive" title="b1">o1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">o2</font> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:18"><span class="lab"><font color="Green" title="E24">A8</font></span></a>, <a class="txt" href="#E16:18"><span class="lab"><font color="Green" title="E29">A13</font></span></a></span>;<br/>
<a NAME="E53:18"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A52</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_6.html#K1" title="CAT_6:func.1">Mor</a> <font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:18"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E54:18"/><span class="lab"><font color="Green" title="E53">A53</font></span>: 
<font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:18"><span class="lab"><font color="Green" title="E45">A33</font></span></a>, <a class="txt" href="#E43:18"><span class="lab"><font color="Green" title="E47">A35</font></span></a>, <a class="ref" href="cat_6.html#D23" onmouseover="rs('cat_6/D23')" onmouseout="rh()">CAT_6:def 23</a></span>;<br/>
<a NAME="E55:18"/><span class="lab"><font color="Green" title="E54">A54</font></span>: 
<font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c4">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:18"><span class="lab"><font color="Green" title="E47">A35</font></span></a>, <a class="txt" href="#E40:18"><span class="lab"><font color="Green" title="E45">A33</font></span></a>, <a class="ref" href="cat_6.html#D23" onmouseover="rs('cat_6/D23')" onmouseout="rh()">CAT_6:def 23</a></span>;<br/>
<a NAME="E56:18"/><span class="lab"><font color="Green" title="E55">A55</font></span>: 
 not <font color="Maroon" title="c6">g2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c4">g1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="18_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:18_8"/>
<font color="Maroon" title="c6">g2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c4">g1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:18_8"/><span class="kw">then </span>
<font color="Maroon" title="c6">g2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c4">g1</font> is   <a href="cat_6.html#NM3" title="CAT_6:NM.3">Object</a> of <font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:18"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="ref" href="cat_6.html#T22" onmouseover="rs('cat_6/T22')" onmouseout="rh()">CAT_6:22</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:18_8"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="txt" href="#E44:18"><span class="lab"><font color="Green" title="E48">A36</font></span></a>, <a class="txt" href="#E42:18"><span class="lab"><font color="Green" title="E46">A34</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E57:18"/><span class="lab"><font color="Green" title="E56">A56</font></span>: 
 not <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="18_9"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:18_9"/><span class="lab"><font color="Green" title="E57">A57</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:18_9"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E53:18"><span class="lab"><font color="Green" title="E52">A52</font></span></a>, <a class="ref" href="cat_6.html#D1" onmouseover="rs('cat_6/D1')" onmouseout="rh()">CAT_6:def 1</a></span>;<br/>
<a NAME="E3:18_9"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T18" onmouseover="rs('funct_1/T18')" onmouseout="rh()">FUNCT_1:18</a></span>;<br/>
<a NAME="E4:18_9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A58</font></span>: 
<span class="p1">(<span class="default"><a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c4">g1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:18"><span class="lab"><font color="Green" title="E49">A37</font></span></a>, <a class="ref" href="struct_0.html#D4" onmouseover="rs('struct_0/D4')" onmouseout="rh()">STRUCT_0:def 4</a></span>;<br/>
<a NAME="E5:18_9"/>
<font color="Maroon" title="c3">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">g2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c4">g1</font></span>)</span></span>)</span> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E55:18"><span class="lab"><font color="Green" title="E54">A54</font></span></a>, <a class="txt" href="#E1:18_9"><span class="lab"><font color="Green" title="E57">A57</font></span></a>, <a class="ref" href="cat_6.html#D22" onmouseover="rs('cat_6/D22')" onmouseout="rh()">CAT_6:def 22</a>, <a class="txt" href="#E8:18"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="ref" href="cat_6.html#D25" onmouseover="rs('cat_6/D25')" onmouseout="rh()">CAT_6:def 25</a></span>;<br/>
<a NAME="E6:18_9"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c4">g1</font></span>)</span> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:18"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="txt" href="#E14:18"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="ref" href="cat_6.html#T34" onmouseover="rs('cat_6/T34')" onmouseout="rh()">CAT_6:34</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:18_9"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E56:18"><span class="lab"><font color="Green" title="E55">A55</font></span></a>, <a class="txt" href="#E45:18"><span class="lab"><font color="Green" title="E49">A37</font></span></a>, <a class="txt" href="#E9:18"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="txt" href="#E14:18"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="txt" href="#E4:18_9"><span class="lab"><font color="Green" title="E58">A58</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E58:18"/>
 <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 3</span>)</span>
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">o11</font> =  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span> as   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 3</span>)</span> ;<br/>
<a NAME="E60:18"/>
 <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 3</span>)</span>
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">o22</font> =  <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span> as   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 3</span>)</span> ;<br/>
<a NAME="E62:18"/>
 <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 3</span>)</span>
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">o33</font> =  <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span> as   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 3</span>)</span> ;<br/>
<a NAME="E64:18"/><span class="lab"><font color="Green" title="E57">A59</font></span>: 
( <font color="Maroon" title="c13">o11</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Maroon" title="c14">o22</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Maroon" title="c15">o33</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#T22" onmouseover="rs('cat_6/T22')" onmouseout="rh()">CAT_6:22</a></span>;<br/>
<a NAME="E65:18"/><span class="lab"><font color="Green" title="E58">A60</font></span>: 
<font color="Maroon" title="c2">F</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_8.html#T9" target="_self" onmouseover="rs('cat_8/T9')" onmouseout="rh()">Th9</a>, <a class="txt" href="#E8:18"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="txt" href="#E9:18"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="ref" href="cat_7.html#D19" onmouseover="rs('cat_7/D19')" onmouseout="rh()">CAT_7:def 19</a></span>;<br/>
<a NAME="E66:18"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E67:18"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A61</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_6.html#K1" title="CAT_6:func.1">Mor</a> <font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D1" onmouseover="rs('cat_6/D1')" onmouseout="rh()">CAT_6:def 1</a></span>;<br/>
<a NAME="E68:18"/><span class="lab"><font color="Green" title="E60">A62</font></span>: 
( <font color="Maroon" title="c10">o0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">o1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c12">o2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c1">C</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18"><span class="lab"><font color="Green" title="E17">A1</font></span></a></span>;<br/>
<a NAME="E69:18"/><span class="lab"><font color="Green" title="E61">A63</font></span>: 
 <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:18"><span class="lab"><font color="Green" title="E41">A29</font></span></a>, <a class="txt" href="#E65:18"><span class="lab"><font color="Green" title="E58">A60</font></span></a>, <a class="txt" href="#E68:18"><span class="lab"><font color="Green" title="E60">A62</font></span></a>, <a class="txt" href="#E67:18"><span class="lab"><font color="Green" title="E59">A61</font></span></a>, <a class="ref" href="funct_1.html#D4" onmouseover="rs('funct_1/D4')" onmouseout="rh()">FUNCT_1:def 4</a></span>;<br/>
<a NAME="E70:18"/><span class="lab"><font color="Green" title="E62">A64</font></span>: 
 <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:18"><span class="lab"><font color="Green" title="E40">A28</font></span></a>, <a class="txt" href="#E38:18"><span class="lab"><font color="Green" title="E43">A31</font></span></a>, <a class="txt" href="#E65:18"><span class="lab"><font color="Green" title="E58">A60</font></span></a>, <a class="txt" href="#E68:18"><span class="lab"><font color="Green" title="E60">A62</font></span></a>, <a class="txt" href="#E67:18"><span class="lab"><font color="Green" title="E59">A61</font></span></a>, <a class="ref" href="funct_1.html#D4" onmouseover="rs('funct_1/D4')" onmouseout="rh()">FUNCT_1:def 4</a>, <a class="txt" href="#E8:18"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="txt" href="#E29:18"><span class="lab"><font color="Green" title="E37">A25</font></span></a>, <a class="ref" href="cat_6.html#T32" onmouseover="rs('cat_6/T32')" onmouseout="rh()">CAT_6:32</a></span>;<br/>
<a NAME="E71:18"/><span class="lab"><font color="Green" title="E63">A65</font></span>: 
 <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:18"><span class="lab"><font color="Green" title="E40">A28</font></span></a>, <a class="txt" href="#E38:18"><span class="lab"><font color="Green" title="E43">A31</font></span></a>, <a class="txt" href="#E65:18"><span class="lab"><font color="Green" title="E58">A60</font></span></a>, <a class="txt" href="#E68:18"><span class="lab"><font color="Green" title="E60">A62</font></span></a>, <a class="txt" href="#E67:18"><span class="lab"><font color="Green" title="E59">A61</font></span></a>, <a class="ref" href="funct_1.html#D4" onmouseover="rs('funct_1/D4')" onmouseout="rh()">FUNCT_1:def 4</a>, <a class="txt" href="#E8:18"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="txt" href="#E29:18"><span class="lab"><font color="Green" title="E37">A25</font></span></a>, <a class="ref" href="cat_6.html#T32" onmouseover="rs('cat_6/T32')" onmouseout="rh()">CAT_6:32</a></span>;<br/>
<a NAME="E72:18"/><span class="lab"><font color="Green" title="E64">A66</font></span>: 
<font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E71:18"><span class="lab"><font color="Green" title="E63">A65</font></span></a>, <a class="txt" href="#E54:18"><span class="lab"><font color="Green" title="E53">A53</font></span></a>, <a class="ref" href="cat_7.html#T4" onmouseover="rs('cat_7/T4')" onmouseout="rh()">CAT_7:4</a></span>;<br/>
<a NAME="E73:18"/>
<font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:18"><span class="lab"><font color="Green" title="E41">A29</font></span></a>, <a class="txt" href="#E65:18"><span class="lab"><font color="Green" title="E58">A60</font></span></a>, <a class="txt" href="#E68:18"><span class="lab"><font color="Green" title="E60">A62</font></span></a>, <a class="txt" href="#E67:18"><span class="lab"><font color="Green" title="E59">A61</font></span></a>, <a class="ref" href="funct_1.html#D4" onmouseover="rs('funct_1/D4')" onmouseout="rh()">FUNCT_1:def 4</a>, <a class="txt" href="#E39:18"><span class="lab"><font color="Green" title="E44">A32</font></span></a>, <a class="txt" href="#E54:18"><span class="lab"><font color="Green" title="E53">A53</font></span></a>, <a class="ref" href="cat_7.html#T4" onmouseover="rs('cat_7/T4')" onmouseout="rh()">CAT_7:4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E74:18"/>
 <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span>, <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span>, <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span>,<font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">g1</font></span>)</span> <a href="zfmisc_1.html#R4" title="ZFMISC_1:pred.4">are_mutually_distinct</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:18"><span class="lab"><font color="Green" title="E35">A19</font></span></a>, <a class="txt" href="#E25:18"><span class="lab"><font color="Green" title="E36">A22</font></span></a>, <a class="txt" href="#E69:18"><span class="lab"><font color="Green" title="E61">A63</font></span></a>, <a class="txt" href="#E57:18"><span class="lab"><font color="Green" title="E56">A56</font></span></a>, <a class="txt" href="#E64:18"><span class="lab"><font color="Green" title="E57">A59</font></span></a>, <a class="txt" href="#E72:18"><span class="lab"><font color="Green" title="E64">A66</font></span></a>, <a class="txt" href="#E70:18"><span class="lab"><font color="Green" title="E62">A64</font></span></a>, <a class="txt" href="#E71:18"><span class="lab"><font color="Green" title="E63">A65</font></span></a>, <a class="ref" href="zfmisc_1.html#D8" onmouseover="rs('zfmisc_1/D8')" onmouseout="rh()">ZFMISC_1:def 8</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
